Problem: Multiply and simplify the following complex numbers: $({1-2i}) \cdot ({4+i})$
Explanation: Complex numbers are multiplied like any two binomials. First use the distributive property: $ ({1-2i}) \cdot ({4+i}) = $ $ ({1} \cdot {4}) + ({1} \cdot {i}) + ({-2i} \cdot {4}) + ({-2i} \cdot {i}) $ Then simplify the terms: $ (4) + (i) + (-8i) + (-2i^2) $ Imaginary unit multiples can be grouped together. $ 4 + (1 - 8)i - 2 i^2 $ After we plug in $i^2 = -1$, the result becomes $ 4 + (1 - 8)i - (-2) $ The result is simplified: $ (4 + 2) + (-7i) = 6-7i $